Example

Let X and R be topological spaces, let $o$ and $t$ be continuous mappings of R into X, let C be the set of pairs $(f, g)\in R^2$ such that $t(f) =o(g)$ in R, and let $m$ be a continuous mapping of C into R making the quiver $(X,R, o, t)$ into a groupoid; suppose moreover that the mapping $f\mapsto f^{-1}$ of R into R is continuous. Then the hypotheses of the proposition are satisfied if one puts $H =\varpi (R)$, $G =\varpi (X),\varphi =\varpi (o),\psi =\varpi (t)$ and $\mu=\varpi (m)$.

Suppose further that R is the graph of an equivalence relation in X, the mappings $o$ and $t$ being deduced from the projections of $X\times X$ into X by passing to the subspaces. Then the hypotheses of the corollary are satisfied.